Used to live in Durham for a while. Durham’s climate is similar to my bee keeping climate where I am in Michigan. You just get a little more snow thanks to the ocean. I do miss some of those big storms.
I have done walk away splits the first week of Aug and had laying queens the first week of September. It takes the queen a while to mate and get laying. The whole process takes about a month. They will lay through September and up until about mid October as you say. A two or three frame split made the first week of August will still only end up about 2 or 3 frames worth of bees come November (as summer bees die). That makes them less winter hardy than 5 or 6 frames of bees.
If you start your splits now, you can probably get to 5 or 6 frames of bees by mid October and those winter better.
I successfully wintered a number of 4 frame medium nucs last winter without putting them over any other hive or adding supplemental heat. I used a well sealed foam bubble over them to do that. In the future, I will probably use supplemental electric heat for anything under 6 medium frames and getting a late start going into winter. Electric heat is a good insurance policy against a cold winter.